Question

In Exercises 73 and \(74,\) determine if any of the planes are parallel or identical. $$ \begin{array}{l} P_{1}: 3 x-2 y+5 z=10 \\ P_{2}:-6 x+4 y-10 z=5 \\ P_{3}:-3 x+2 y+5 z=8 \\ P_{4}: 75 x-50 y+125 z=250 \end{array} $$

Short answer

\(P_{1}\) and \(P_{3}\) are parallel, \(P_{1}\) and \(P_{4}\) are identical, and \(P_{2}\) is neither parallel nor identical to any of the other planes.

Step-by-step solution

Identifying Normal Vectors

The normal vector of a plane is given by the coefficients of the x, y, and z terms. Identify the normal vectors for each plane: for \(P_{1}\), the normal vector is \(<3,-2,5>\), for \(P_{2}\), it's \(<-6,4,-10>\), for \(P_{3}\), it's \(<-3,2,-5>\) and for \(P_{4}\), it's \(<75,-50,125>\).

Checking for Parallel or Identical Planes

Now that we've identified the normal vectors for each plane, we can compare them. Parallel or identical planes will have normal vectors that are scalar multiples of each other. Here, \(P_{1}\) and \(P_{3}\) have the same normal vector up to sign, but the constants on the right side of the equations differ, so they are parallel. Similarly, \(P_{1}\) and \(P_{4}\) have normal vectors that are scalar multiples of each other and the constants on the right side of the equation are also multiplied by the same scalar, so they are identical. \(P_{2}\) is neither parallel nor identical to any of the other planes.

Conclusion

By comparing the normal vectors of the plane equations and their constants, we conclude that \(P_{1}\) and \(P_{3}\) are parallel, \(P_{1}\) and \(P_{4}\) are identical, and \(P_{2}\) is neither parallel nor identical to any of the other planes.